Counting Bounded Tree Depth Homomorphisms [article]

Martin Grohe
2020 arXiv   pre-print
We prove that graphs G, G' satisfy the same sentences of first-order logic with counting of quantifier rank at most k if and only if they are homomorphism-indistinguishable over the class of all graphs of tree depth at most k. Here G, G' are homomorphism-indistinguishable over a class C of graphs if for each graph F in C, the number of homomorphisms from F to G equals the number of homomorphisms from F to G'.
arXiv:2003.08164v1 fatcat:xr3tgpnvmbd55dzq6bj7goqwcm